If a grinding wheel, for example, is used for cutting peripheral grooves into a pipe or other generally cylindrical workpiece rotatable on a suitable holder, or if the workpiece is to be severed by such a tool, a hydraulically or otherwise actuated tool support may be rapidly displaced to advance the tool from a withdrawn position into a workpiece-engaging position, the advance being then continued at a reduced rate which in conventional systems is preselected to take into account not only the composition and the wall thickness of the workpiece but also any possible eccentricity thereof to cause variations of the depth of penetration of the tool as the workpiece rotates about its axis. In order to prevent overheating and possible stoppage, especially at points where the tool first encounters the workpiece, that preselected rate of advance must be slower than is necessary at other points where the workpiece is thinner, softer or farther away from the starting position of the tool.